Most of the trigonometry on the Math IC test addresses
the different parts of a right triangle and the relationships among
these different parts. The three basic trigonometric functions—sine,
cosine, and tangent—are the tools that define these relationships.
Given the measure of one of the non-right angles in a right triangle,
you can use these tools of trigonometry to find the characteristics
of the triangle. If you are given the measure of one of the non-right
angles and one of the sides, you can find all the values of the
right triangle.

Basic Functions and the Right Triangle

If you know the measure of one of the non-right angles
in a right triangle, the trigonometric functions tell you the ratio
of the lengths of any two sides of the triangle.

In the right triangle below, one acute angle is labeled and the sides of the triangle are labeled
hypotenuse, opposite, and adjacent, according to their position
relative to the angle of measure .

Sine

The sine of an angle is the ratio of the
side opposite the angle to the hypotenuse.

Cosine

The cosine of an angle is the ratio of the
side adjacent the angle to the hypotenuse.

Tangent

The tangent of an angle is the ratio of the
side opposite the angle to the side adjacent to the angle.

A handy way to remember these formulas is the acronym
SOHCAHTOA. The S, C, and T stand
for the three different basic trigonometric functions, and the two
letters after the S, C, and T refer
to the sides of the triangle that are being related by that function.

SOH:

Sine is the side Opposite
the angle divided by the Hypotenuse.

CAH:

Cosine is the side Adjacent
to the angle divided by the Hypotenuse.

TOA:

Tangent is the side Opposite
divided by the Adjacent side.

Using Your Calculator with the Basic Functions

On some questions dealing with sine, cosine, and tangent,
your calculator can be extremely helpful. Using your calculator,
you can quickly compute the value of one of the three trigonometric
functions at any given angle. On a graphing calculator, find the
button indicating the trigonometric function you want to perform,
type in the value of the angle, and then hit Enter. To calculate
the cosine of 45º, press the COS button, then type in 45, and press
Enter.

On non-graphing calculators you may need to type in the
value of the angle first and then press the trigonometric-function
button.

Angles Larger than 90º and the Basic Functions

Angles in a right triangle can never be larger than 90º,
since the sum of all three angles must equal 180º. But on the Math
IC, you may occasionally run into angles that are larger than 90º.
It is often more intuitive to think of these in terms of the coordinate
plane rather than in terms of a triangle.

Below are pictured four angles in the coordinate plane.
The first is the acute angle we’ve already covered in this chapter;
the next three are all larger than 90º.

The four quadrants of the coordinate plane become very
important when dealing with angles that are larger than 90º. Each
angle larger than 90º can be “simplified” by looking at it in the
context of its own quadrant. In the figure below, the four angles
from the previous angle are defined in terms of their own quadrants:

By reconsidering each angle based on its relationship
to the x-axis, it becomes clear that each of the
original angles can be treated as a reoriented 30º angle. In other
words, a 210º angle is just the same as a 30º angle except that
the 210º angle is in the third quadrant. In terms of the basic trigonometric
functions, this means that the value of a 210º angle is the same
as the value of a 30º angle, except that the sign of the trigonometric
function differs based on the quadrant that the angle is in. Depending
on the quadrant of the coordinate plane in which an angle resides,
the values of the trigonometric properties of that angle will be
either positive or negative. Below is a figure illustrating the
signs of the trigonometric functions according to the quadrant in
which they lie.

You should memorize the above graph.

The Math IC will probably test whether you know the proper
sign for each quadrant in an indirect way, meaning that it’s unlikely
that you’ll have to do any heavy calculating when dealing with this
topic. Instead, you might find a question such as:

If
the value of sin – is .5, what is the value
of sin?

This question doesn’t ask you to think about sin for any
specific value of . Instead, it tests your
understanding of the quadrant signs for the sine function. The first
thing you should notice is that – and have the same magnitude, even if
they have different signs. This means that the magnitude of sine
for – and will be the same. Immediately you
should understand that sin must equal either .5 or –.5. To
figure out which of these values is right, you have to decide what
quadrant angle resides in. Based
on the graph of the sine function or from the above chart, you can
see that the sine function has a positive value in quadrants I and
II, and negative values in quadrants III and IV. Since sin – is equal to a positive number, .5,
you know that – must represent
an angle in quadrant I or II. Since angle is simply the reflection of – across the x-axis,
you can see that angle must be in either
quadrant III or IV. The value of sin must be negative: –.5 is the right
answer.